Projective spectrum of a ring

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A scheme associated with a graded ring (cf. also Graded module). As a set of points, is a set of homogeneous prime ideals such that does not contain . The topology on is defined by the following basis of open sets: for , . The structure sheaf of the locally ringed space is defined on the basis open sets as follows: , that is, the subring of the elements of degree of the ring of fractions with respect to the multiplicative system .

The most important example of a projective spectrum is . The set of its -valued points for any field is in natural correspondence with the set of points of the -dimensional projective space over the field .

If all the rings as -modules are spanned by ( terms), then an additional structure is defined on . Namely, the covering and the units determine a Čech -cocycle on to which an invertible sheaf, denoted by , corresponds. The symbol usually denotes the -th tensor power of . There exists a canonical homomorphism , indicating the geometric meaning of the grading of the ring (see [1]). If, for example, , then corresponds to a sheaf of hyperplane sections in .


[1] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701
[2] A. Grothendieck, "Eléments de géometrie algebrique" Publ. Math. IHES , 1–4 (1960–1967) MR0238860 MR0217086 MR0199181 MR0173675 MR0163911 MR0217085 MR0217084 MR0163910 MR0163909 MR0217083 MR0163908 Zbl 0203.23301 Zbl 0144.19904 Zbl 0135.39701 Zbl 0136.15901 Zbl 0118.36206


See also Projective scheme.


[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Projective spectrum of a ring. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article